Whereas much more sophisticated schemes for vertical turbulent diffusion may spontaneously simulate some kind of vertical convection (see for example, Mellor and Yamada (1974)), our simple scheme (as many others) is not able to prevent sub-adiabatic vertical temperature gradients
If such an unstable profile is produced by the model, an adiabatic profile is immediately restored with a simple energy conserving scheme. If the resulting temperature profile is unstable at its upper or lower limit this mechanism is instantaneously extended in such a way that the final profile is entirely stable.
This convective adjustment is in fact achieved in a real atmosphere
by parcel exchange through vertical convective motions.
These motions not only transport energy but also momentum.
The intensity of the momentum exchange is linked to the mass fluxes
involved in the convection, which cannot be estimated in such a
simple model.
The best one can do is to compute
an estimate of the instability of the atmosphere from the
relative enthalpy exchange necessary to restore the adiabatic profile
from the original profile
When (this condition is always verified in the
simulations) the angular momentum is entirely mixed on a fraction
of the mesh. This rather arbitrary choice is qualitatively
acceptable in the sense that a larger instability will produce
a larger momentum mixing.